A systematic search for conformal field theories in very small spaces

ArXiv.org · 2025-09-04 · 1 citations

Groundstates of 1+1d conformal field theories (CFTs) satisfy a local entropic condition called the vector fixed point equation. This condition is surprisingly well satisfied by groundstates of quantum critical lattice models even at small system sizes. We perform a search in the space of states of very small systems (four qubits and four qutrits) and examine the states that satisfy this condition. By reconstructing a local Hamiltonian from each state, we are able to identify many of these solutions with known CFTs; others are gapped fixed points, or involve large relevant perturbations, and others are CFTs we have not yet identified. These ideas are also useful for identifying continuous quantum phase transitions in a given family of Hamiltonians, and for identifying the nature of the critical theory in small systems.

A critical theory for solidification of a liquid Fermi liquid

ArXiv.org · 2025-04-09

We give a simple description of a zero-temperature phase transition between a liquid metal and a solid. The critical point has a Fermi surface as well as a Bose surface, a sphere in momentum space of gapless bosonic excitations. We find a fixed point of the renormalization group governing such a non-Fermi liquid, using an expansion in the codimension of both the Fermi and Bose surfaces. We comment on the nature of the solid phase and possible physical realizations.

Strict Area Law Entanglement versus Chirality

Physical Review Letters · 2025-05-06 · 1 citations

Chirality is a property of a gapped phase of matter in two spatial dimensions that can be manifested through nonzero thermal or electrical Hall conductance. In this Letter, we prove two no-go theorems that forbid such chirality for a quantum state in a finite dimensional local Hilbert space with strict area law entanglement entropies. We also show that the finite dimensional local Hilbert space condition can be relaxed to the condition that the state has finite local entanglement entropies. As a crucial ingredient in the proofs, we introduce a new quantum information-theoretic primitive called instantaneous modular flow, which has many other potential applications.

Conformal geometry from entanglement

SciPost Physics · 2025-03-19 · 3 citations

In a physical system with conformal symmetry, observables depend on cross-ratios, measures of distance invariant under global conformal transformations (conformal geometry for short). We identify a quantum information-theoretic mechanism by which the conformal geometry emerges at the gapless edge of a 2+1D quantum many-body system with a bulk energy gap. We introduce a novel pair of information-theoretic quantities (\mathfrak{c}_{\textrm{tot}}, \eta) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msub> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔠</mml:mi> </mml:mstyle> <mml:mtext mathvariant="normal">tot</mml:mtext> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>η</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> that can be defined locally on the edge from the wavefunction of the many-body system, without prior knowledge of any distance measure. We posit that, for a topological groundstate, the quantity \mathfrak{c}_{\textrm{tot}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔠</mml:mi> </mml:mstyle> <mml:mtext mathvariant="normal">tot</mml:mtext> </mml:msub> </mml:math> is stationary under arbitrary variations of the quantum state, and study the logical consequences. We show that stationarity, modulo an entanglement-based assumption about the bulk, implies (i) \mathfrak{c}_{\textrm{tot}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔠</mml:mi> </mml:mstyle> <mml:mtext mathvariant="normal">tot</mml:mtext> </mml:msub> </mml:math> is a non-negative constant that can be interpreted as the total central charge of the edge theory. (ii) \eta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>η</mml:mi> </mml:math> is a cross-ratio, obeying the full set of mathematical consistency rules, which further indicates the existence of a distance measure of the edge with global conformal invariance. Thus, the conformal geometry emerges from a simple assumption on groundstate entanglement. We show that stationarity of \mathfrak{c}_{\textrm{tot}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔠</mml:mi> </mml:mstyle> <mml:mtext mathvariant="normal">tot</mml:mtext> </mml:msub> </mml:math> is equivalent to a vector fixed-point equation involving \eta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>η</mml:mi> </mml:math> , making our assumption locally checkable. We also derive similar results for 1+1D systems under a suitable set of assumptions.

Remote detectability from entanglement bootstrap I: Kirby’s torus trick

SciPost Physics · 2025-04-14 · 1 citations

Remote detectability is often taken as a physical assumption in the study of topologically ordered systems, and it is a central axiom of mathematical frameworks of topological quantum field theories. We show under the entanglement bootstrap approach that remote detectability is a necessary property; that is, we derive it as a theorem. Starting from a single wave function on a topologically-trivial region satisfying the entanglement bootstrap axioms, we can construct states on closed manifolds. The crucial technique is to immerse the punctured manifold into the topologically trivial region and then heal the puncture. This is analogous to Kirby’s torus trick. We then analyze a special class of such manifolds, which we call pairing manifolds. For each pairing manifold, which pairs two classes of excitations, we identify an analog of the topological S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>S</mml:mi> </mml:math> -matrix. This pairing matrix is unitary, which implies remote detectability between two classes of excitations. These matrices are in general not associated with the mapping class group of the manifold. As a by-product, we can count excitation types (e.g., graph excitations in 3+1d). The pairing phenomenon occurs in many physical contexts, including systems in different dimensions, with or without gapped boundaries. We provide a variety of examples to illustrate its scope.

Nishimori's self-tuning as evidence for the existence of God

arXiv (Cornell University) · 2024-04-01

Apparent violations of Naturalness may be explained by positing the existence of an omniscient but disinterested and possibly fallible Observer who regularly performs von Neumann measurements on us (and everything else). We comment briefly on the implications for the construction of scalable quantum computers.

Chiral Virasoro algebra from a single wavefunction

Annals of Physics · 2024-11-16 · 3 citations

Chiral edges of 2+1D systems can have very robust emergent conformal symmetry. When the edge is purely chiral, the Hilbert space of low-energy edge excitations can form a representation of a single Virasoro algebra. We propose a method to systematically extract the generators of the Virasoro algebra from a single ground state wavefunction, using entanglement bootstrap and an input from the edge conformal field theory. We corroborate our construction by numerically verifying the commutation relations of the generators. We also study the unitary flows generated by these operators, whose properties (such as energy and state overlap) are shown numerically to agree with our analytical predictions.

Dimer piling problems and interacting field theory

Physical review. D/Physical review. D. · 2024-09-23

The dimer tiling problem asks in how many ways can the edges of a graph be covered by dimers so that each site is covered once. In the special case of a planar graph, this problem has a solution in terms of a free fermionic field theory. We rediscover and explore an expression for the number of coverings of an arbitrary graph by arbitrary objects in terms of an interacting fermionic field theory first proposed by Samuel. Generalizations of the dimer tiling problem, which we call “dimer piling problems,” demand that each site be covered <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi></a:math> times by indistinguishable dimers. Our field theory provides a solution of these problems in the large-<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>N</c:mi></c:math> limit. We give a similar path integral representation for certain lattice coloring problems. Published by the American Physical Society 2024

Strict area law entanglement versus chirality

arXiv (Cornell University) · 2024-08-19 · 1 citations

Chirality is a property of a gapped phase of matter in two spatial dimensions that can be manifested through non-zero thermal or electrical Hall conductance. In this paper, we prove two no-go theorems that forbid such chirality for a quantum state in a finite dimensional local Hilbert space with strict area law entanglement entropies. As a crucial ingredient in the proofs, we introduce a new quantum information-theoretic primitive called instantaneous modular flow, which has many other potential applications.

Conformal geometry from entanglement

arXiv (Cornell University) · 2024-04-04 · 2 citations

In a physical system with conformal symmetry, observables depend on cross-ratios, measures of distance invariant under global conformal transformations (conformal geometry for short). We identify a quantum information-theoretic mechanism by which the conformal geometry emerges at the gapless edge of a 2+1D quantum many-body system with a bulk energy gap. We introduce a novel pair of information-theoretic quantities $(\mathfrak{c}_{\mathrm{tot}}, η)$ that can be defined locally on the edge from the wavefunction of the many-body system, without prior knowledge of any distance measure. We posit that, for a topological groundstate, the quantity $\mathfrak{c}_{\mathrm{tot}}$ is stationary under arbitrary variations of the quantum state, and study the logical consequences. We show that stationarity, modulo an entanglement-based assumption about the bulk, implies (i) $\mathfrak{c}_{\mathrm{tot}}$ is a non-negative constant that can be interpreted as the total central charge of the edge theory. (ii) $η$ is a cross-ratio, obeying the full set of mathematical consistency rules, which further indicates the existence of a distance measure of the edge with global conformal invariance. Thus, the conformal geometry emerges from a simple assumption on groundstate entanglement. We show that stationarity of $\mathfrak{c}_{\mathrm{tot}}$ is equivalent to a vector fixed-point equation involving $η$, making our assumption locally checkable. We also derive similar results for 1+1D systems under a suitable set of assumptions.